Solve 27^n=10

$$27 ^ { n } = 10$$

Solve for n

$$n=\frac{\log_{3}\left(10\right)}{3}\approx 0.698634425$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$27^{n}=10$$

Take the logarithm of both sides of the equation.

$$\log(27^{n})=\log(10)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$n\log(27)=\log(10)$$

Divide both sides by \log(27).

$$n=\frac{\log(10)}{\log(27)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$n=\log_{27}\left(10\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{3\ln(3)}+\frac{\log_{3}\left(10\right)}{3}$$

$$n_{1}\in \mathrm{Z}$$